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Differentiability
Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.
\[\lambda=\left( \frac{f(5102)-f(2015)}{f'(c)} \right) \left( \frac{f^2(2015)+f^2(5102)+f(2015)f(5102)}{f^2(c)} \right)\]
Let \(f:[2015,5102] \rightarrow [0,\infty)\)be any continuous and differentiable function. Find the value of \(\lambda\), such that there exists some \(c\in [2015,5102]\) which satisfies the equation above.
by
Sandeep Bhardwaj
\[ f(x) = \displaystyle \prod_{k=1}^{\infty} \left( \dfrac{1+ 2\cos\left( \dfrac{2x}{3^k} \right)}{3}\right) \]
Let \(f(x)\) denote an function as described above. The number of points where \( |xf(x)| + | |x-2|-1| \) is non-differentiable in \( x \in (0,3\pi) \) is \(k\), find \(k^2\).
by
Upanshu Gupta
such that the tangents at these points to \(S\) are perpendicular to the line \[l \ : \ (\sqrt {2}-1)x+y=0.\]
Find the value of \[\left\lfloor \displaystyle\sum_{k=1}^{n} |\alpha_k|-|\beta_k|\right\rfloor.\]
Details and Assumptions
\(\lfloor{\cdots}\rfloor\) denotes the floor function (greatest integer function).
Even though in the problem, a plural form of points is given, there may only be one such point that exists.
by
Pratik Shastri
What is the largest possible number of integers \(n\) such that for some fixed polynomial \(f(x) \) of degree \(3\) with integer coefficients, \(f(n)=n^{1000}\)?
by
Calvin Lin
Let \(f(x)\) be a non-constant thrice differentiable function defined on real numbers such that \(f(x)=f(6-x)\) and \(f'(0)=0=f'(2)=f'(5)\). Find the minimum number of values of \(p \in [0,6]\) which satisfy the equation \[(f''(p))^2+f'(p)f'''(p)=0\] Details and Assumptions:
\(f'(p)=\left( \frac{df(x)}{dx} \right)_{x=p}\)
\(f''(p)=\left( \frac{d^2f(x)}{dx^2} \right)_{x=p}\)
\(f'''(p)=\left( \frac{d^3f(x)}{dx^3} \right)_{x=p}\)
by
Sandeep Bhardwaj